Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
21 (2005), 55–61
www.emis.de/journals
ISSN 1786-0091
NEIGHBORHOODS OF A CLASS OF ANALYTIC FUNCTIONS
WITH NEGATIVE COEFFICIENTS
HALIT ORHAN & MUHAMMET KAMALI
Abstract. The main object of this paper is to prove several inclusion relations
associated with the (n, δ) neighborhoods of various subclasses of starlike and
convex functions of complex order by making use of the known concept of
neighborhoods of analytic functions. Special cases of some of these inclusion
relations are shown to yield known results.
1. Introduction
Let A(n) denote the class of functions f (z) of the form
(1.1)
f (z) = z −
∞
X
ak z k ,
(ak ≥ 0;
n ∈ N := {1, 2, 3, . . .}),
k=n+1
which are analytic in the open unit disk
∆ = {z : z ∈ C,
|z| < 1}.
Following [6, 9], we defined the (n, δ)-neighborhood of a function f (z) ∈ A(n) by
(
)
∞
∞
X
X
(1.2) Nn,δ (f ) := g ∈A(n): g(z) =z−
bk z k and
k |ak − bk | ≤ δ .
k=n+1
k=n+1
In particular, for the identity function
(1.3)
e(z) = z,
we immediately have
(
(1.4)
Nn,δ (e) :=
g ∈A(n): g(z) = z −
∞
X
k=n+1
bk z k and
∞
X
)
k |bk | ≤ δ
.
k=n+1
A function f (z) ∈ A(n) is said to be starlike of complex order γ(γ ∈ C − {0}),
that is, f ∈ Sn∗ (γ), if it also satisfies the inequality
¾
½
1 zf 0 (z)
−1) > 0 (z ∈ ∆; γ ∈ C − {0}).
(1.5)
Re 1 + (
γ f (z)
2000 Mathematics Subject Classification. 30C45.
Key words and phrases. Analytic functions, starlike functions, convex functions, (n, δ)neighborhood, inclusion relations, identity function.
55
56
HALIT ORHAN & MUHAMMET KAMALI
Furthermore, a function f (z) ∈ A(n) is said to be convex of complex order γ
(γ ∈ C − {0}), that is, f ∈ Cn (γ), if it also satisfies the inequality
½
¾
1 zf 0 (z)
(1.6)
Re 1 + (
) > 0, (z ∈ ∆; γ ∈ C − {0}).
γ f (z)
The classes Sn∗ (γ) and Cn (γ) stem essentially from the classes of starlike and
convex functions of complex order, which were considered earlier by Nasr and Aouf
[8] and Wiatrowski [13], respectively, (see also [5, 11]).
Let Sn (γ, λ, β) denote the subclass of A(n) consisting of functions f (z) which
satisfy the inequality
¯ ½
¾¯
¯
¯1
zf 0 (z) + λz 2 f 00 (z)
¯
¯
¯ γ λzf 0 (z) + (1 − λ)f (z) − 1 ¯ < β,
(z ∈ ∆; γ ∈ C − {0} ; 0 ≤ λ ≤ 1; 0 < β ≤ 1).
Let <n (γ, λ, β) denote the subclass of A(n) consisting of functions f (z) which
satisfy the inequality
¯
¯
¯1 0
¯
¯ {f (z) + λzf 00 (z) − 1}¯ < β,
¯γ
¯
(z ∈ ∆; γ ∈ C − {0} ; 0 ≤ λ ≤ 1; 0 < β ≤ 1).
Neighborhoods of the classes Sn (γ, λ, β) and <n (γ, λ, β) investigated by Altıntaş,
Özkan and Srivastava [3].
∞
P
Let A be class of functions f (z) of the form f (z) = z +
ak z k which are
k=2
analytic in the open unit disk ∆ = {z : |z| < 1}. For f (z) belong to A, Salagean
[10] has introduced the following operator called the Salagean operator:
D0 f (z) = f (z), D1 f (z) = Df (z) = zf 0 (z)
Dn f (z) = D(Dn−1 f (z)) (n ∈ N = {1, 2, 3, . . .}).
Note that Dn f (z) = z +
∞
P
k n ak z k , n∈ N0 = N ∪ {0}.
k=2
Let A(n) denote the class of functions f (z) of the form
(1.7)
f (z) = z −
∞
X
ak+1 z k+1 ,
(ak ≥ 0; n ∈ N := {1, 2, 3, . . .})
k=n
which analytic in the open unit disk
∆ = {z : z ∈ C, |z| < 1} .
We define the (n, δ)-neighborhood of a function f (z) ∈ A(n) by
(1.8)
)
(
∞
∞
X
X
k+1
bk+1 z
∧
(k + 1) |ak+1 − bk+1 | ≤ δ .
Nn,δ (f ) := g ∈ ∆(n) : g(z) = z −
k=n
In particular, for the identity function
e(z) = z,
k=n
NEIGHBORHOODS OF A CLASS OF ANALYTIC FUNCTIONS. . .
we immediately have
(
(1.9) Nn,δ (e) :=
g ∈ A(n) : g(z) = z −
∞
X
bk+1 z
k+1
k=n
∧
∞
X
57
)
(k + 1) |bk+1 | ≤ δ
.
k=n
We can write the following equalities for the functions f (z) belong to the class
A(n)
D0 f (z) = f (z),
D1 f (z) = Df (z) = zf 0 (z) = z −
∞
X
(k + 1)ak+1 z k+1 ,
k=n
D2 f (z) = D(Df (z)) = z −
∞
X
(k + 1)2 ak+1 z k+1 ,
k=n
...
DΩ f (z) = D(DΩ−1 f (z)) = z −
∞
X
(k + 1)Ω ak+1 z k+1 , (Ω ∈ N ∪ {0}) [7].
k=n
Let Sn (γ, λ, β, Ω) denote the subclass of A(n)consisting of functions f (z) which
satisfy the inequality
¯ µ
¶¯
¯
¯ 1 (1 − λ)z(DΩ f (z))0 + λz(DΩ+1 f (z))0
¯
− 1 ¯¯ < β
¯γ
(1 − λ)DΩ f (z) + λDΩ+1 f (z)
(1.10)
(z ∈ ∆; γ ∈ C − {0} ; 0 ≤ λ ≤ 1; 0 < β ≤ 1; Ω ∈ N ∪ {0}).
Also let <n (γ, λ, β, Ω) denote the subclass of A(n) consisting of functions f (z)
which satisfy the inequality
¯
¯
¯1 ¡
¢¯
¯ (1 − λ)(DΩ f (z))0 + λ(DΩ+1 f (z))0 − 1 ¯ < β,
¯
¯γ
(1.11)
(z ∈ ∆; γ ∈ C − {0} ; 0 ≤ λ ≤ 1; 0 < β ≤ 1; Ω ∈ N ∪ {0}).
Various further subclasses of the classes Sn (γ, λ, β, 0) and <n (γ, λ, β, 0) with
γ = 1 were studied in many earlier works (cf., e.g., [1, 12] ; see also the references
cited in these earlier works). Clearly, we have
Sn (γ, 0, 1, 0) ⊂ Sn∗ (γ) and <n (γ, 0, 1, 0) ⊂ Cn (γ),
(n ∈ N, γ ∈ C − {0}).
The main object of the present paper is to investigate the (n, δ)-neighborhoods
of the following Sn (γ, λ, β, Ω) and <n (γ, λ, β, Ω) subclasses of the class A(n) of
normalized analytic functions in ∆ with negative coefficients.
2. A set of inclusion relations involving Nn,δ (e)
In our investigation of the inclusion relations involving Nn,δ (e), we need the
following Lemma 1 and Lemma 2.
Lemma 1. Let the function f ∈ A(n) be defined by (1.7), then f (z) is in the class
Sn (γ, λ, β, Ω) if and only if
(2.1)
∞
X
k=n
(k + 1)Ω (λk + 1)(k + β |γ|)ak+1 ≤ β |γ|
58
HALIT ORHAN & MUHAMMET KAMALI
Proof. We suppose that f ∈ Sn (γ, λ, β, Ω). Then, by appealing to condition (1.10),
we readily get
½
¾
(1 − λ)z(DΩ f (z))0 + λz(DΩ+1 f (z))0
(2.2)
Re
−
1
> −β |γ|
(1 − λ)DΩ f (z) + λDΩ+1 f (z)
or equivalently,


∞
P
Ω
k+1 



−
(k
+
1)
k(λk
+
1)a
z
k+1


k=n
(2.3)
Re
> −β |γ| , (z ∈ ∆)
∞
P



(k + 1)Ω (λk + 1)ak+1 z k+1 
z −

k=n
where we have made use of definition (1.7).
Now choose values of z on the real axis and let z → 1− through real values.
Then inequality (2.3) immediately yields the wanted condition (2.1).
Conversely, by applying hypothesis (2.1) and letting |z| = 1, we find that
¯
¯
¯
¯ (1 − λ)z(DΩ f (z))0 + λz(DΩ+1 f (z))0
¯
¯
−
1
¯
¯
Ω
Ω+1
(1 − λ)D f (z) + λD
f (z)
¯ ∞
¯
¯ P
¯
¯
(k + 1)Ω k(λk + 1)ak+1 z k+1 ¯
¯ k=n
¯
¯
= ¯¯
∞
¯
¯ z − P (k + 1)Ω (λk + 1)ak+1 z k+1 ¯
¯
¯
(2.4)
k=n
½
¾
∞
P
Ω
β |γ| 1 −
(k + 1) (λk + 1)ak+1
k=n
≤
∞
P
1−
(k + 1)Ω (λk + 1)ak+1
k=n
≤ β |γ| .
Hence, by maximum modulus theorem, we have f ∈ Sn (γ, λ, β, Ω), which evidently
completes the proof of lemma 1.
¤
Similarly, we can prove the following.
Lemma 2. Let the function f ∈ A(n) be defined by (1.7), then f (z) is in the class
<n (γ, λ, β, Ω) if and only if
∞
X
(2.5)
(k + 1)Ω+1 (λk + 1)ak+1 ≤ β |γ|.
k=n
Remark 1. A special case of Lemma 1 when γ = 1, Ω = 0 and β = 1 − α
(0 ≤ α < 1) was given earlier by Altıntaş [1, p. 489, Theorem 1].
Our first inclusion relation involving Nn,δ (e) is given by the following.
Theorem 1. Let
(2.6)
δ=
(n +
β |γ|
+ 1)(n + β |γ|)
1)Ω−1 (λn
then
(2.7)
Sn (γ, λ, β, Ω) ⊂ Nn,δ (e).
(|γ| < 1),
NEIGHBORHOODS OF A CLASS OF ANALYTIC FUNCTIONS. . .
59
Proof. For f ∈ Sn (γ, λ, β, Ω), lemma 1 immediately yields
∞
X
(n + 1)Ω (λn + 1)(n + β |γ|)
ak+1 ≤ β |γ|
k=n
so that
∞
X
(2.8)
ak+1 ≤
k=n
(n +
β |γ|
.
+ 1)(n + β |γ|)
1)Ω (λn
On the other hand, we also find from (2.1) and (2.8) that
∞
X
(k + 1)Ω (λk + 1)(k + β |γ|)ak+1 ≤ β |γ|
k=n
⇒
∞
X
(k + 1)Ω (λk + 1)(k + 1 − 1 + β |γ|)ak+1 ≤ β |γ|
k=n
⇒ (n + 1)Ω (λn + 1)
∞
X
(k + 1)ak+1
k=n
≤ β |γ| + (1 − β |γ|)(n + 1)Ω (λn + 1)
∞
X
ak+1
k=n
≤ β |γ| + (1 − β |γ|)(n + 1)Ω (λn + 1)
(n +
β |γ|
(n + 1)β |γ|
= β |γ| + (1 − β |γ|)
=
.
n + β |γ|
n + β |γ|
Thus
∞
X
(k + 1)ak+1 ≤
k=n
(n +
β |γ|
+ 1)(n + β |γ|)
1)Ω (λn
β |γ|
+ 1)(n + β |γ|)
1)Ω−1 (λn
(|γ| < 1),
that is
(2.9)
∞
X
k=n
(k + 1)ak+1 ≤
(n +
β |γ|
= δ,
+ 1)(n + β |γ|)
1)Ω−1 (λn
which, in view of definition (1.9), proves Theorem 1.
¤
By similarly, applying Lemma 2 instead of Lemma 1, we can prove the following.
Theorem 2. Let
(2.10)
δ=
β |γ|
(λn + 1)(n + 1)Ω
then
(2.11)
<n (γ, λ, β, Ω) ⊂ Nn,δ (e).
Remark 2. A special case of Theorem 1 and Theorem 2 when Ω = 0 was proven
recently by Altıntaş, Özkan and Srivastava [3].
Remark 3. A special case of Theorem 1 when
(2.12)
γ = 1 − α, (0 ≤ α < 1), λ = Ω = 0
and
β=1
was proven recently by Altıntaş and Owa [2, p. 798, Theorem 2.1].
60
HALIT ORHAN & MUHAMMET KAMALI
(α)
(α)
3. Neighborhoods for the classes Sn (γ, λ, β, Ω) and Rn (γ, λ, β, Ω)
In this section, we determine the neighborhood for each of the classes
Sn(α) (γ, λ, β, Ω) and <(α)
n (γ, λ, β, Ω),
which we define as follows. A function f ∈ A(n) be defined by (1.7) is said to be
(α)
in the class Sn (γ, λ, β, Ω) if there exist a function g ∈ Sn (γ, λ, β, Ω) such that
¯
¯
¯ f (z)
¯
¯
¯
(3.1)
¯ g(z) − 1¯ < 1 − α, (z ∈ ∆; 0 ≤ α < 1).
Analogously, a function f ∈ A(n) be defined by (1.7) is said to be in the class
(α)
<n (γ, λ, β, Ω) if there exists a function g ∈ <n (γ, λ, β, Ω) such that inequality
(3.1) holds true.
Theorem 3. If g ∈ Sn (γ, λ, β, Ω) and
(3.2)
α=1−
δ(n + 1)Ω−1 (λn + 1)(n + β |γ|)
(n + 1)Ω (λn + 1)(n + β |γ|) − β |γ|
then
Nn,δ (g) ⊂ Sn(α) (γ, λ, β, Ω).
(3.3)
Proof. Suppose that f ∈ Nn,δ (g). We then find from (1.8) that
∞
X
(3.4)
(k + 1) |ak+1 − bk+1 | ≤ δ
k=n
which readily implies the coefficients inequality
(3.5)
∞
X
|ak+1 − bk+1 | ≤
k=n
δ
,
n+1
(n ∈ N).
Next, since g ∈ Sn (γ, λ, β, Ω), we have (cf. Equation (2.8))
(3.6)
∞
X
bk+1 ≤
k=n
so that
(n +
β |γ|
,
+ 1)(n + β |γ|)
1)Ω (λn
¯
¯ P∞
¯ f (z)
¯
¯
¯ < k=n |ak+1 − bk+1 |
−
1
∞
¯ g(z)
¯
P
1−
bk+1
k=n
(3.7)
δ
.
≤
n+1 1−
1
β|γ|
(n+1)Ω (λn+1)(n+β|γ|)
δ(n + 1)Ω−1 (λn + 1)(n + β |γ|)
(n + 1)Ω (λn + 1)(n + β |γ|) − β |γ|
= 1 − α,
=
(α)
provided that α is given precisely by (3.2). Thus, by definition, f ∈ Sn (γ, λ, β, Ω)
for α given by (3.2), which evidently completes our proof of Theorem 3.
¤
Our proof of Theorem 4 below is much similar to that of Theorem 3.
NEIGHBORHOODS OF A CLASS OF ANALYTIC FUNCTIONS. . .
61
Theorem 4. If g ∈ <n (γ, λ, β, Ω) and
(3.8)
α=1−
δ(n + 1)Ω (λn + 1)
(n + 1)Ω+1 (λn + 1) − β |γ|
then
(3.9)
Nn,δ (g) ⊂ <(α)
n (γ, λ, β, Ω).
Remark 4. A special case of Theorem 3 and Theorem 4 when Ω = 0 was proven
recently by Altıntaş, Özkan and Srivastava [3].
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Received October 21, 2003; revised Apryl 28, 2004.
Department of Mathematics,
Faculty of Science and Art,
Atatürk University,
25240 Erzurum, Turkey
E-mail address: horhan@atauni.edu.tr